How can I control multiple points of a cloth sim?
Instead of constraints you could hook the left extremity to an empty and add the vertices of the opposite extremity to the pin group. Put the Hook modifier above the Cloth modifier. In the Physics properties, increase the Quality Steps up to 20 to make it work correctly. You don't tell about what's supposed to happen after the last picture of your gif, does the extremity on the right stay still?


Even simpler, you could give up cloth simulation and use a Curve modifier: